pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot, ggpubr, spdep, GWmodel, tmap, skimr, caret)In-Class Exercise 5
1 Overview
In this In-Class Exercise, we will demonstrate the basic concepts and methods of logistic regression specially designed for geographical data. In particular, we will demonstrate the following:
explain the similarities and differences between Logistic Regression (LR) algorithm versus geographical weighted Logistic Regression (GWLR) algorithm.
calibrate predictive models by using appropriate Geographically Weighted Logistic Regression algorithm for geographical data.
2 The Data
In this exercise, we will analyse the data from Nigeria. There are 2 datasets used, as outlined in sections 2.1 and 2.2. We will have chosen Osun for this analysis as this state has a relatively high proportion of non-functional water points compared to the other states in Nigeria.
2.1 Aspatial Data
Data was downloaded from WPdx Global Data Repositories in a csv format. The WPdx+ data set was filtered for “nigeria” in the column clean_country_name before downloading. There is a total of 95,008 unique water point records.
2.2 Geospatial Data
Nigeria Level-2 Administrative Boundary (also known as Local Government Area, LGA) polygon features GIS data was downloaded from geoBoundaries.
3 Getting Started
The R packages needed for this exercise are as follows:
4 Importing the Analytical Data
In the following code chunk, we will import Osun.rds and Osun_wp_sf.rds that have been previously tidied by using read_rds(). In Osun.rds, we have kept the geographical boundary for Osun state to allow for better plotting later.
Osun <- read_rds("rds/Osun.rds")
Osun_wp_sf <- read_rds("rds/Osun_wp_sf.rds")We check our independent variable i.e. status by running the following code chunk. We can see that it is a binary data - TRUE representing functional water points and FALSE representing non-functional water points. We can see that there are 55.5% functional water points and 44.5% non-functional water points.
Osun_wp_sf %>%
freq(input = 'status')Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
Please report the issue at <https://github.com/pablo14/funModeling/issues>.
status frequency percentage cumulative_perc
1 TRUE 2642 55.5 55.5
2 FALSE 2118 44.5 100.0tmap_mode("view")tmap mode set to interactive viewingtm_shape(Osun)+
tm_polygons(alpha=0.4)+
tm_shape(Osun_wp_sf)+
tm_dots(col="status",
alpha=0.6)+
tm_view(set.zoom.limits = c(9,12))5 Exploratory Data Analysis
Regression models are very sensitive to excessive number of missing values (e.g. fields with more than 20-50% missing values, depending on the sample size). In this section, we will look at distribution of the variables. We will use skimr() which will allow the results to be displayed in a nice report.
Osun_wp_sf %>%
skim()Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.| Name | Piped data |
| Number of rows | 4760 |
| Number of columns | 75 |
| _______________________ | |
| Column type frequency: | |
| character | 47 |
| logical | 5 |
| numeric | 23 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| source | 0 | 1.00 | 5 | 44 | 0 | 2 | 0 |
| report_date | 0 | 1.00 | 22 | 22 | 0 | 42 | 0 |
| status_id | 0 | 1.00 | 2 | 7 | 0 | 3 | 0 |
| water_source_clean | 0 | 1.00 | 8 | 22 | 0 | 3 | 0 |
| water_source_category | 0 | 1.00 | 4 | 6 | 0 | 2 | 0 |
| water_tech_clean | 24 | 0.99 | 9 | 23 | 0 | 3 | 0 |
| water_tech_category | 24 | 0.99 | 9 | 15 | 0 | 2 | 0 |
| facility_type | 0 | 1.00 | 8 | 8 | 0 | 1 | 0 |
| clean_country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| clean_adm1 | 0 | 1.00 | 3 | 5 | 0 | 5 | 0 |
| clean_adm2 | 0 | 1.00 | 3 | 14 | 0 | 35 | 0 |
| clean_adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| clean_adm4 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| installer | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management_clean | 1573 | 0.67 | 5 | 37 | 0 | 7 | 0 |
| status_clean | 0 | 1.00 | 9 | 32 | 0 | 7 | 0 |
| pay | 0 | 1.00 | 2 | 39 | 0 | 7 | 0 |
| fecal_coliform_presence | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| subjective_quality | 0 | 1.00 | 18 | 20 | 0 | 4 | 0 |
| activity_id | 4757 | 0.00 | 36 | 36 | 0 | 3 | 0 |
| scheme_id | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| wpdx_id | 0 | 1.00 | 12 | 12 | 0 | 4760 | 0 |
| notes | 0 | 1.00 | 2 | 96 | 0 | 3502 | 0 |
| orig_lnk | 4757 | 0.00 | 84 | 84 | 0 | 1 | 0 |
| photo_lnk | 41 | 0.99 | 84 | 84 | 0 | 4719 | 0 |
| country_id | 0 | 1.00 | 2 | 2 | 0 | 1 | 0 |
| data_lnk | 0 | 1.00 | 79 | 96 | 0 | 2 | 0 |
| water_point_history | 0 | 1.00 | 142 | 834 | 0 | 4750 | 0 |
| clean_country_id | 0 | 1.00 | 3 | 3 | 0 | 1 | 0 |
| country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| water_source | 0 | 1.00 | 8 | 30 | 0 | 4 | 0 |
| water_tech | 0 | 1.00 | 5 | 37 | 0 | 20 | 0 |
| adm2 | 0 | 1.00 | 3 | 14 | 0 | 33 | 0 |
| adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management | 1573 | 0.67 | 5 | 47 | 0 | 7 | 0 |
| adm1 | 0 | 1.00 | 4 | 5 | 0 | 4 | 0 |
| New Georeferenced Column | 0 | 1.00 | 16 | 35 | 0 | 4760 | 0 |
| lat_lon_deg | 0 | 1.00 | 13 | 32 | 0 | 4760 | 0 |
| public_data_source | 0 | 1.00 | 84 | 102 | 0 | 2 | 0 |
| converted | 0 | 1.00 | 53 | 53 | 0 | 1 | 0 |
| created_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| updated_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| Geometry | 0 | 1.00 | 33 | 37 | 0 | 4760 | 0 |
| ADM2_EN | 0 | 1.00 | 3 | 14 | 0 | 30 | 0 |
| ADM2_PCODE | 0 | 1.00 | 8 | 8 | 0 | 30 | 0 |
| ADM1_EN | 0 | 1.00 | 4 | 4 | 0 | 1 | 0 |
| ADM1_PCODE | 0 | 1.00 | 5 | 5 | 0 | 1 | 0 |
Variable type: logical
| skim_variable | n_missing | complete_rate | mean | count |
|---|---|---|---|---|
| rehab_year | 4760 | 0 | NaN | : |
| rehabilitator | 4760 | 0 | NaN | : |
| is_urban | 0 | 1 | 0.39 | FAL: 2884, TRU: 1876 |
| latest_record | 0 | 1 | 1.00 | TRU: 4760 |
| status | 0 | 1 | 0.56 | TRU: 2642, FAL: 2118 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| row_id | 0 | 1.00 | 68550.48 | 10216.94 | 49601.00 | 66874.75 | 68244.50 | 69562.25 | 471319.00 | ▇▁▁▁▁ |
| lat_deg | 0 | 1.00 | 7.68 | 0.22 | 7.06 | 7.51 | 7.71 | 7.88 | 8.06 | ▁▂▇▇▇ |
| lon_deg | 0 | 1.00 | 4.54 | 0.21 | 4.08 | 4.36 | 4.56 | 4.71 | 5.06 | ▃▆▇▇▂ |
| install_year | 1144 | 0.76 | 2008.63 | 6.04 | 1917.00 | 2006.00 | 2010.00 | 2013.00 | 2015.00 | ▁▁▁▁▇ |
| fecal_coliform_value | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| distance_to_primary_road | 0 | 1.00 | 5021.53 | 5648.34 | 0.01 | 719.36 | 2972.78 | 7314.73 | 26909.86 | ▇▂▁▁▁ |
| distance_to_secondary_road | 0 | 1.00 | 3750.47 | 3938.63 | 0.15 | 460.90 | 2554.25 | 5791.94 | 19559.48 | ▇▃▁▁▁ |
| distance_to_tertiary_road | 0 | 1.00 | 1259.28 | 1680.04 | 0.02 | 121.25 | 521.77 | 1834.42 | 10966.27 | ▇▂▁▁▁ |
| distance_to_city | 0 | 1.00 | 16663.99 | 10960.82 | 53.05 | 7930.75 | 15030.41 | 24255.75 | 47934.34 | ▇▇▆▃▁ |
| distance_to_town | 0 | 1.00 | 16726.59 | 12452.65 | 30.00 | 6876.92 | 12204.53 | 27739.46 | 44020.64 | ▇▅▃▃▂ |
| rehab_priority | 2654 | 0.44 | 489.33 | 1658.81 | 0.00 | 7.00 | 91.50 | 376.25 | 29697.00 | ▇▁▁▁▁ |
| water_point_population | 4 | 1.00 | 513.58 | 1458.92 | 0.00 | 14.00 | 119.00 | 433.25 | 29697.00 | ▇▁▁▁▁ |
| local_population_1km | 4 | 1.00 | 2727.16 | 4189.46 | 0.00 | 176.00 | 1032.00 | 3717.00 | 36118.00 | ▇▁▁▁▁ |
| crucialness_score | 798 | 0.83 | 0.26 | 0.28 | 0.00 | 0.07 | 0.15 | 0.35 | 1.00 | ▇▃▁▁▁ |
| pressure_score | 798 | 0.83 | 1.46 | 4.16 | 0.00 | 0.12 | 0.41 | 1.24 | 93.69 | ▇▁▁▁▁ |
| usage_capacity | 0 | 1.00 | 560.74 | 338.46 | 300.00 | 300.00 | 300.00 | 1000.00 | 1000.00 | ▇▁▁▁▅ |
| days_since_report | 0 | 1.00 | 2692.69 | 41.92 | 1483.00 | 2688.00 | 2693.00 | 2700.00 | 4645.00 | ▁▇▁▁▁ |
| staleness_score | 0 | 1.00 | 42.80 | 0.58 | 23.13 | 42.70 | 42.79 | 42.86 | 62.66 | ▁▁▇▁▁ |
| location_id | 0 | 1.00 | 235865.49 | 6657.60 | 23741.00 | 230638.75 | 236199.50 | 240061.25 | 267454.00 | ▁▁▁▁▇ |
| cluster_size | 0 | 1.00 | 1.05 | 0.25 | 1.00 | 1.00 | 1.00 | 1.00 | 4.00 | ▇▁▁▁▁ |
| lat_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| lon_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| count | 0 | 1.00 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | ▁▁▇▁▁ |
We can also see the number of missing values in each field. For instance, install_year has 1144 missing values out of a total of 4760 records for Osun. In this case, we cannot use this field (24% missing values) although it is a useful variable since we know water points that are beyond 8-9 years are more likely to be non-functional.
On the other hand, we can see that fields local_population_1km and water_point_population both have 4 missing values, which is a small number of records, and hence we can still use these 2 fields.
In the following code chunk, we will select the independent variables that we will use for our regression model and to exclude records that have missing values. We will use as.factor() for usage_capacity as there are only specific values for the capacity of the water points, hence, we should not treat this field as a continuous variables and instead, make the values as factor.
Osun_wp_sf_clean <- Osun_wp_sf %>%
filter_at(vars(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
is_urban,
water_source_clean),
all_vars(!is.na(.))) %>%
mutate(usage_capacity = as.factor(usage_capacity))6 Correlation Analysis
In this section, we want to know if any of the numerical independent variables are correlated. We will first need to drop the geometry column in the spatial data Osun_wp_sf_clean so that the geometry field does not interfere with correlation analysis.
Osun_wp <- Osun_wp_sf_clean %>%
select(c(7,35:39,42:43,46:47,57)) %>%
st_set_geometry(NULL)We can then perform correlation analysis only on the numerical variables.
cluster_vars.cor = cor(
Osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
lower = "ellipse",
upper = "number",
tl.pos = "lt",
diag = "l",
tl.col = "black")
We can see that there is no multicollinearity observed among the numerical variables (no coefficient greater than 0.8).
7 Building a Logistic Regression Model
In the following code check, we will use glm of R to calibrate a logistic regression model for the water point status.
model <- glm(status ~ distance_to_primary_road+
distance_to_secondary_road+
distance_to_tertiary_road+
distance_to_city+
distance_to_town+
is_urban+
usage_capacity+
water_source_clean+
water_point_population+
local_population_1km,
data = Osun_wp_sf_clean,
family = binomial(link = 'logit'))In the results for model, we can see that fitted.values are all probability values which is our y-hat.
We will then use blr_regress() from blorr package to generate a report for the model results.
blr_regress(model) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4744 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3887 0.1124 3.4588 5e-04
distance_to_primary_road 1 0.0000 0.0000 -0.7153 0.4744
distance_to_secondary_road 1 0.0000 0.0000 -0.5530 0.5802
distance_to_tertiary_road 1 1e-04 0.0000 4.6708 0.0000
distance_to_city 1 0.0000 0.0000 -4.7574 0.0000
distance_to_town 1 0.0000 0.0000 -4.9170 0.0000
is_urbanTRUE 1 -0.2971 0.0819 -3.6294 3e-04
usage_capacity1000 1 -0.6230 0.0697 -8.9366 0.0000
water_source_cleanProtected Shallow Well 1 0.5040 0.0857 5.8783 0.0000
water_source_cleanProtected Spring 1 1.2882 0.4388 2.9359 0.0033
water_point_population 1 -5e-04 0.0000 -11.3686 0.0000
local_population_1km 1 3e-04 0.0000 19.2953 0.0000
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7347 Somers' D 0.4693
% Discordant 0.2653 Gamma 0.4693
% Tied 0.0000 Tau-a 0.2318
Pairs 5585188 c 0.7347
---------------------------------------------------------------We can see that distance_to_primary_road and distance_to_secondary_road has p-value greater than 0.05, we will subsequently exclude these 2 fields which are not statistically significant (i.e. p-value < 0.05). For categorical variables, a positive value indicates an average correlation and a negative value implies a below average correlation.
For continuous variables, a positive value implies a direct correlation and a negative value implies an inverse relation, while the magnitude of the coefficient represents the strength of the correlations. We will make this analysis for the continuous variables after we have confirmed that they are statistically significant.
In the code chunk below, blr_confusion_matrix() from blorr package to prepare a confusion matrix. We will use cutoff = 0.5, this means that if the fitted.values sf greater than 0.5, we will label the water point as functional, and if the fitted.values determined is less than 0.5, we will label the water point as non-functional. (The validity of the cutoff is measured using accuracy, sensitivity, and specificity).
blr_confusion_matrix(model, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1301 738
1 813 1904
Accuracy : 0.6739
No Information Rate : 0.4445
Kappa : 0.3373
McNemars's Test P-Value : 0.0602
Sensitivity : 0.7207
Specificity : 0.6154
Pos Pred Value : 0.7008
Neg Pred Value : 0.6381
Prevalence : 0.5555
Detection Rate : 0.4003
Detection Prevalence : 0.5713
Balanced Accuracy : 0.6680
Precision : 0.7008
Recall : 0.7207
'Positive' Class : 1We can also see that our accruracy is approximately 67.4%. The sensitivity is higher than the specificity, indicating that our true positive is higher (correctly determined approximately 72% true positive) than the true negative (model correctly determines approximately 62% true negative).
8 Building Geographically Weighted Logistic Regression (gwLR) Models
8.1 Converting from sf to sp data frame
We will use select() from dplyr package to select the variables fo interest. We will convert the data to SpatialPointsDataFrame data type for compatibility with subsequent packages. We will need to use Osun_wp_sf_clean which excludes the 4 polygons with missing values as polygons with missing values will cause an error prompt.
Osun_wp_sp <- Osun_wp_sf_clean %>%
select(c(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
is_urban,
usage_capacity,
water_source_clean)) %>%
as_Spatial()
Osun_wp_spclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 11
names : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean
min values : 0, 0.014461356813335, 0.152195902540837, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 0, 1000, Borehole
max values : 1, 26909.8616132094, 19559.4793799085, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 1, 300, Protected Spring Since our geometry data is already in projected coordinate format, we can set the longlat as FALSE (the following result will match the SI unit of the projected coordinate system). We will set the argument adaptive to FALSE which indicates that we are interested to compute the fixed bandwidth. We will leave all variables including distance_to_primary_road and distance_to_secondary_road in the following code chunk.
8.2 Building Fixed Bandwidth GWR Model
We can plot a basic gwlr using the bandwidth obtained earlier.
8.2.1 Computing Fixed Bandwidth
bw.fixed <- bw.ggwr(status ~ distance_to_primary_road+
distance_to_secondary_road+
distance_to_tertiary_road+
distance_to_city+
distance_to_town+
is_urban+
usage_capacity+
water_source_clean+
water_point_population+
local_population_1km,
data = Osun_wp_sp,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2889
1 -2836
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5684.357
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2875
1 -2818
2 -2810
3 -2808
4 -2808
5 -2808
Fixed bandwidth: 59200.13 AICc value: 5646.785
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2847
1 -2781
2 -2768
3 -2765
4 -2765
5 -2765
6 -2765
Fixed bandwidth: 36599.53 AICc value: 5575.148
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2798
1 -2719
2 -2698
3 -2693
4 -2693
5 -2693
6 -2693
Fixed bandwidth: 22631.59 AICc value: 5466.883
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2720
1 -2622
2 -2590
3 -2581
4 -2580
5 -2580
6 -2580
7 -2580
Fixed bandwidth: 13998.93 AICc value: 5324.578
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2601
1 -2476
2 -2431
3 -2419
4 -2417
5 -2417
6 -2417
7 -2417
Fixed bandwidth: 8663.649 AICc value: 5163.61
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2436
1 -2268
2 -2194
3 -2167
4 -2161
5 -2161
6 -2161
7 -2161
8 -2161
9 -2161
Fixed bandwidth: 5366.266 AICc value: 4990.587
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2157
1 -1922
2 -1802
3 -1739
4 -1713
5 -1713
Fixed bandwidth: 3328.371 AICc value: 4798.288
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1751
1 -1421
2 -1238
3 -1133
4 -1084
5 -1084
Fixed bandwidth: 2068.882 AICc value: 4837.017
Iteration Log-Likelihood:(With bandwidth: 4106.777 )
=========================
0 -2297
1 -2095
2 -1997
3 -1951
4 -1938
5 -1936
6 -1936
7 -1936
8 -1936
Fixed bandwidth: 4106.777 AICc value: 4873.161
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2036
1 -1771
2 -1633
3 -1558
4 -1525
5 -1525
Fixed bandwidth: 2847.289 AICc value: 4768.192
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -1941
1 -1655
2 -1503
3 -1417
4 -1378
5 -1378
Fixed bandwidth: 2549.964 AICc value: 4762.212
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1874
1 -1573
2 -1410
3 -1316
4 -1274
5 -1274
Fixed bandwidth: 2366.207 AICc value: 4773.081
Iteration Log-Likelihood:(With bandwidth: 2663.532 )
=========================
0 -1979
1 -1702
2 -1555
3 -1474
4 -1438
5 -1438
Fixed bandwidth: 2663.532 AICc value: 4762.568
Iteration Log-Likelihood:(With bandwidth: 2479.775 )
=========================
0 -1917
1 -1625
2 -1468
3 -1380
4 -1339
5 -1339
Fixed bandwidth: 2479.775 AICc value: 4764.294
Iteration Log-Likelihood:(With bandwidth: 2593.343 )
=========================
0 -1956
1 -1674
2 -1523
3 -1439
4 -1401
5 -1401
Fixed bandwidth: 2593.343 AICc value: 4761.813
Iteration Log-Likelihood:(With bandwidth: 2620.153 )
=========================
0 -1965
1 -1685
2 -1536
3 -1453
4 -1415
5 -1415
Fixed bandwidth: 2620.153 AICc value: 4761.89
Iteration Log-Likelihood:(With bandwidth: 2576.774 )
=========================
0 -1950
1 -1667
2 -1515
3 -1431
4 -1393
5 -1393
Fixed bandwidth: 2576.774 AICc value: 4761.889
Iteration Log-Likelihood:(With bandwidth: 2603.584 )
=========================
0 -1960
1 -1678
2 -1528
3 -1445
4 -1407
5 -1407
Fixed bandwidth: 2603.584 AICc value: 4761.813
Iteration Log-Likelihood:(With bandwidth: 2609.913 )
=========================
0 -1962
1 -1680
2 -1531
3 -1448
4 -1410
5 -1410
Fixed bandwidth: 2609.913 AICc value: 4761.831
Iteration Log-Likelihood:(With bandwidth: 2599.672 )
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405
Fixed bandwidth: 2599.672 AICc value: 4761.809
Iteration Log-Likelihood:(With bandwidth: 2597.255 )
=========================
0 -1957
1 -1675
2 -1525
3 -1441
4 -1403
5 -1403
Fixed bandwidth: 2597.255 AICc value: 4761.809 bw.fixed[1] 2599.672We obtain a fixed bandwidth of 2599.672 m (projection in Nigeria is in metres), which is approximately 2.6 km.
8.2.2 Building fixed bandwidth model
gwlr.fixed <- ggwr.basic(status ~ distance_to_primary_road+
distance_to_secondary_road+
distance_to_tertiary_road+
distance_to_city+
distance_to_town+
is_urban+
usage_capacity+
water_source_clean+
water_point_population+
local_population_1km,
data = Osun_wp_sp,
bw = bw.fixed,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405 gwlr.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-18 00:03:15
Call:
ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road +
distance_to_tertiary_road + distance_to_city + distance_to_town +
is_urban + usage_capacity + water_source_clean + water_point_population +
local_population_1km, data = Osun_wp_sp, bw = bw.fixed, family = "binomial",
kernel = "gaussian", adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-124.555 -1.755 1.072 1.742 34.333
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.887e-01 1.124e-01 3.459 0.000543
distance_to_primary_road -4.642e-06 6.490e-06 -0.715 0.474422
distance_to_secondary_road -5.143e-06 9.299e-06 -0.553 0.580230
distance_to_tertiary_road 9.683e-05 2.073e-05 4.671 3.00e-06
distance_to_city -1.686e-05 3.544e-06 -4.757 1.96e-06
distance_to_town -1.480e-05 3.009e-06 -4.917 8.79e-07
is_urbanTRUE -2.971e-01 8.185e-02 -3.629 0.000284
usage_capacity1000 -6.230e-01 6.972e-02 -8.937 < 2e-16
water_source_cleanProtected Shallow Well 5.040e-01 8.574e-02 5.878 4.14e-09
water_source_cleanProtected Spring 1.288e+00 4.388e-01 2.936 0.003325
water_point_population -5.097e-04 4.484e-05 -11.369 < 2e-16
local_population_1km 3.451e-04 1.788e-05 19.295 < 2e-16
Intercept ***
distance_to_primary_road
distance_to_secondary_road
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
is_urbanTRUE ***
usage_capacity1000 ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
water_point_population ***
local_population_1km ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.0 on 4744 degrees of freedom
AIC: 5712
Number of Fisher Scoring iterations: 5
AICc: 5712.099
Pseudo R-square value: 0.1295351
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2599.672
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -8.7228e+02 -4.9955e+00 1.7600e+00
distance_to_primary_road -1.9389e-02 -4.8031e-04 2.9618e-05
distance_to_secondary_road -1.5921e-02 -3.7551e-04 1.2317e-04
distance_to_tertiary_road -1.5618e-02 -4.2368e-04 7.6179e-05
distance_to_city -1.8416e-02 -5.6217e-04 -1.2726e-04
distance_to_town -2.2411e-02 -5.7283e-04 -1.5155e-04
is_urbanTRUE -1.9790e+02 -4.2908e+00 -1.6864e+00
usage_capacity1000 -2.0772e+01 -9.7231e-01 -4.1592e-01
water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01 5.3340e-01
water_source_cleanProtected.Spring -5.2235e+02 -5.5977e+00 2.5441e+00
water_point_population -5.2208e-02 -2.2767e-03 -9.8875e-04
local_population_1km -1.2698e-01 4.9952e-04 1.0638e-03
3rd Qu. Max.
Intercept 1.2763e+01 1073.2154
distance_to_primary_road 4.8443e-04 0.0142
distance_to_secondary_road 6.0692e-04 0.0258
distance_to_tertiary_road 6.6814e-04 0.0128
distance_to_city 2.3718e-04 0.0150
distance_to_town 1.9271e-04 0.0224
is_urbanTRUE 1.2841e+00 744.3097
usage_capacity1000 3.0322e-01 5.9281
water_source_cleanProtected.Shallow.Well 1.7849e+00 67.6343
water_source_cleanProtected.Spring 6.7663e+00 317.4123
water_point_population 5.0102e-04 0.1309
local_population_1km 1.8157e-03 0.0392
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2795.084
AIC : 4414.606
AICc : 4747.423
Pseudo R-square value: 0.5722559
***********************************************************************
Program stops at: 2022-12-18 00:04:11 From the results, we can see that the Geographically Weighted Regression model has a lower AIC compared to the Generalised Linear Regression. We cannot use the AICc because the global model (Generalised Linear Regression, which does not have geographical information) does not have AICc. This tells us that Geographically Weighted Regression model has improved explainability.
8.3 Model Assessment
8.3.1 Converting SDF into sf data.frame
To assess the model, we will first convert the model into SFD object as data.frame using the following code chunk.
gwr.fixed <- as.data.frame(gwlr.fixed$SDF)Next, we will label the yhat values (i.e. predicted probability) greater or equal to 0.5 into 1 and else 0. The result of the logic comparison operation will be saved into a field called most.
gwr.fixed <- gwr.fixed %>%
mutate(most = ifelse(
gwr.fixed$yhat >= 0.5, T, F))We will use confusionMatrix() from caret package to generate the confusion matrix. We define data argument to be the predicted probability and reference argument to be the actual label (i.e. ground truth).
gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most, reference = gwr.fixed$y)
CMConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1824 263
TRUE 290 2379
Accuracy : 0.8837
95% CI : (0.8743, 0.8927)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7642
Mcnemar's Test P-Value : 0.2689
Sensitivity : 0.8628
Specificity : 0.9005
Pos Pred Value : 0.8740
Neg Pred Value : 0.8913
Prevalence : 0.4445
Detection Rate : 0.3835
Detection Prevalence : 0.4388
Balanced Accuracy : 0.8816
'Positive' Class : FALSE
When we compare the overall accuracy is now improved to 88.37% (geographically weighted) compared to the 67.39% that we obtained initially in the non-geographically weighted. In addition, the sensitivity improved from 72% to 86%. Also specificity improved from 62% to 90%. This implies we should apply a local strategy (looking at surrounding neighbours) instead of a global strategy to understand the factors for water points being functional or non-functional.
(Note that for the global model, you can see the coefficients of each independent variable. But we do not see this for the local geographically weighted model because one model is built for each state, and hence there are over 4000+ of such coefficients for each variable).
Osun_wp_sf_selected <- Osun_wp_sf_clean %>%
select(c(ADM2_EN, ADM2_PCODE,
ADM1_EN, ADM1_PCODE,
status))gwr_sf.fixed <- cbind(Osun_wp_sf_selected, gwr.fixed)8.4 Visualising gwLR
The following code chunk below is used to create an interactive point symbol map to compare the actual status of the water points against the status of the water points predicted by gwLR.
tmap_mode("view")tmap mode set to interactive viewingprob_T <- tm_shape(Osun)+
tm_polygons(alpha = 0.1)+
tm_shape(gwr_sf.fixed)+
tm_dots(col = "yhat",
border.col = "gray60",
border.lwd = 1)+
tm_view(set.zoom.limits = c(8,14))
prob_TIn the following, we will visualise how the standard error of coefficient and t-value for the field distance_to_tertiary_road differs for the local models obtained for each LGA in Osun.
The standard error of the coefficient measures how precisely the model estimates the coefficient’s unknown value. The standard error of the coefficient is always positive.
The smaller the standard error, the more precise the estimate. Dividing the coefficient by its standard error calculates a t-value.
tertiary_TV <- tm_shape(Osun)+
tm_polygons(alpha=0.1)+
tm_shape(gwr_sf.fixed)+
tm_dots(col="distance_to_tertiary_road_TV",
border.col="gray60",
border.lwd = 1)+
tm_view(set.zoom.limits = c(8,14))
tertiary_SE <- tm_shape(Osun)+
tm_polygons(alpha=0.1)+
tm_shape(gwr_sf.fixed)+
tm_dots(col="distance_to_tertiary_road_SE",
border.col="gray60",
border.lwd = 1)+
tm_view(set.zoom.limits = c(8,14))
tmap_arrange(tertiary_SE, tertiary_TV, asp=1, ncol=2, sync=TRUE)Variable(s) "distance_to_tertiary_road_TV" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.With reference to the plot on the left, we can see that the standard error of coefficient for distance_to_tertiary_road is generally low for all areas (as indicated by the yellow dots), with exception of dots in red colour, i.e. LGAs in Atakumosa East.
9 Building Logistic Regression and Geographically Weighted Logistic Regression Models using only Statistically Significant Independent Variables
In this section, we will only use independent variables that are statistically significant. Like before, we will build both logistic regression model and geographically weighted logistic regression model and then compare the 2 models.
9.1 Logistic regression model
In the following code check, we will use glm of R to calibrate a logistic regression model for the water point status. We will exclude fields that are not statistically significant, i.e. distance_to_primary_road and distance_to_secondary_road.
model_sig <- glm(status ~ distance_to_tertiary_road+
distance_to_city+
distance_to_town+
is_urban+
usage_capacity+
water_source_clean+
water_point_population+
local_population_1km,
data = Osun_wp_sf_clean,
family = binomial(link = 'logit'))
blr_regress(model_sig) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4746 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3540 0.1055 3.3541 8e-04
distance_to_tertiary_road 1 1e-04 0.0000 4.9096 0.0000
distance_to_city 1 0.0000 0.0000 -5.2022 0.0000
distance_to_town 1 0.0000 0.0000 -5.4660 0.0000
is_urbanTRUE 1 -0.2667 0.0747 -3.5690 4e-04
usage_capacity1000 1 -0.6206 0.0697 -8.9081 0.0000
water_source_cleanProtected Shallow Well 1 0.4947 0.0850 5.8228 0.0000
water_source_cleanProtected Spring 1 1.2790 0.4384 2.9174 0.0035
water_point_population 1 -5e-04 0.0000 -11.3902 0.0000
local_population_1km 1 3e-04 0.0000 19.4069 0.0000
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7349 Somers' D 0.4697
% Discordant 0.2651 Gamma 0.4697
% Tied 0.0000 Tau-a 0.2320
Pairs 5585188 c 0.7349
---------------------------------------------------------------We can see that all independent variables used to build this model are statistically significant (i.e. p-value < 0.05).
blr_confusion_matrix(model_sig, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1300 743
1 814 1899
Accuracy : 0.6726
No Information Rate : 0.4445
Kappa : 0.3348
McNemars's Test P-Value : 0.0761
Sensitivity : 0.7188
Specificity : 0.6149
Pos Pred Value : 0.7000
Neg Pred Value : 0.6363
Prevalence : 0.5555
Detection Rate : 0.3993
Detection Prevalence : 0.5704
Balanced Accuracy : 0.6669
Precision : 0.7000
Recall : 0.7188
'Positive' Class : 1We can also see that our accuracy is approximately 67%. The sensitivity is also higher than the specificity, indicating that our true positive is higher (model has correctly determined approximately 72% true positive) than the true negative (model correctly determines approximately 61% true negative).
9.2 Geographically weighted logistic regression model (with fixed bandwidth)
9.2.1 Converting from sf to sp data frame
We will first convert the data to a SpatialPointsDataFrame data type for compatibility with subsequent packages. In here, we have also excluded fields that are not statistically significant, i.e. distance_to_primary_road and distance_to_secondary_road.
Osun_wp_sp_sig <- Osun_wp_sf_clean %>%
select(c(status,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
is_urban,
usage_capacity,
water_source_clean,
water_point_population,
local_population_1km)) %>%
as_Spatial()
Osun_wp_sp_sigclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 9
names : status, distance_to_tertiary_road, distance_to_city, distance_to_town, is_urban, usage_capacity, water_source_clean, water_point_population, local_population_1km
min values : 0, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 1000, Borehole, 0, 0
max values : 1, 10966.2705628969, 47934.343603562, 44020.6393368124, 1, 300, Protected Spring, 29697, 36118 9.2.2 Computing fixed bandwidth
Here, we will compute the fixed bandwidth that we will use to build the geographically weighted logistic regression model.
bw.fixed.sig <- bw.ggwr(status ~ distance_to_tertiary_road+
distance_to_city+
distance_to_town+
is_urban+
usage_capacity+
water_source_clean+
water_point_population+
local_population_1km,
data = Osun_wp_sp_sig,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2890
1 -2837
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5681.18
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2878
1 -2820
2 -2812
3 -2810
4 -2810
5 -2810
Fixed bandwidth: 59200.13 AICc value: 5645.901
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2854
1 -2790
2 -2777
3 -2774
4 -2774
5 -2774
6 -2774
Fixed bandwidth: 36599.53 AICc value: 5585.354
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2810
1 -2732
2 -2711
3 -2707
4 -2707
5 -2707
6 -2707
Fixed bandwidth: 22631.59 AICc value: 5481.877
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2732
1 -2635
2 -2604
3 -2597
4 -2596
5 -2596
6 -2596
Fixed bandwidth: 13998.93 AICc value: 5333.718
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2624
1 -2502
2 -2459
3 -2447
4 -2446
5 -2446
6 -2446
7 -2446
Fixed bandwidth: 8663.649 AICc value: 5178.493
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2478
1 -2319
2 -2250
3 -2225
4 -2219
5 -2219
6 -2220
7 -2220
8 -2220
9 -2220
Fixed bandwidth: 5366.266 AICc value: 5022.016
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2222
1 -2002
2 -1894
3 -1838
4 -1818
5 -1814
6 -1814
Fixed bandwidth: 3328.371 AICc value: 4827.587
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1837
1 -1528
2 -1357
3 -1261
4 -1222
5 -1222
Fixed bandwidth: 2068.882 AICc value: 4772.046
Iteration Log-Likelihood:(With bandwidth: 1290.476 )
=========================
0 -1403
1 -1016
2 -807.3
3 -680.2
4 -680.2
Fixed bandwidth: 1290.476 AICc value: 5809.719
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -2019
1 -1753
2 -1614
3 -1538
4 -1506
5 -1506
Fixed bandwidth: 2549.964 AICc value: 4764.056
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2108
1 -1862
2 -1736
3 -1670
4 -1644
5 -1644
Fixed bandwidth: 2847.289 AICc value: 4791.834
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1955
1 -1675
2 -1525
3 -1441
4 -1407
5 -1407
Fixed bandwidth: 2366.207 AICc value: 4755.524
Iteration Log-Likelihood:(With bandwidth: 2252.639 )
=========================
0 -1913
1 -1623
2 -1465
3 -1376
4 -1341
5 -1341
Fixed bandwidth: 2252.639 AICc value: 4759.188
Iteration Log-Likelihood:(With bandwidth: 2436.396 )
=========================
0 -1980
1 -1706
2 -1560
3 -1479
4 -1446
5 -1446
Fixed bandwidth: 2436.396 AICc value: 4756.675
Iteration Log-Likelihood:(With bandwidth: 2322.828 )
=========================
0 -1940
1 -1656
2 -1503
3 -1417
4 -1382
5 -1382
Fixed bandwidth: 2322.828 AICc value: 4756.471
Iteration Log-Likelihood:(With bandwidth: 2393.017 )
=========================
0 -1965
1 -1687
2 -1539
3 -1456
4 -1422
5 -1422
Fixed bandwidth: 2393.017 AICc value: 4755.57
Iteration Log-Likelihood:(With bandwidth: 2349.638 )
=========================
0 -1949
1 -1668
2 -1517
3 -1432
4 -1398
5 -1398
Fixed bandwidth: 2349.638 AICc value: 4755.753
Iteration Log-Likelihood:(With bandwidth: 2376.448 )
=========================
0 -1959
1 -1680
2 -1530
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2376.448 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2382.777 )
=========================
0 -1961
1 -1683
2 -1534
3 -1450
4 -1416
5 -1416
Fixed bandwidth: 2382.777 AICc value: 4755.491
Iteration Log-Likelihood:(With bandwidth: 2372.536 )
=========================
0 -1958
1 -1678
2 -1528
3 -1445
4 -1411
5 -1411
Fixed bandwidth: 2372.536 AICc value: 4755.488
Iteration Log-Likelihood:(With bandwidth: 2378.865 )
=========================
0 -1960
1 -1681
2 -1532
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2378.865 AICc value: 4755.481
Iteration Log-Likelihood:(With bandwidth: 2374.954 )
=========================
0 -1959
1 -1679
2 -1530
3 -1446
4 -1412
5 -1412
Fixed bandwidth: 2374.954 AICc value: 4755.482
Iteration Log-Likelihood:(With bandwidth: 2377.371 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.371 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.942 )
=========================
0 -1960
1 -1680
2 -1531
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2377.942 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.018 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.018 AICc value: 4755.48 bw.fixed.sig[1] 2377.371We obtain a fixed bandwidth of 2377.371 m (projection in Nigeria is in metres), which is approximately 2.4 km.
9.2.3 Building fixed bandwidth model
Next, we will build a geographically weighted logistic regression model using the fixed bandwidth determined in the previous section.
gwlr.fixed.sig <- ggwr.basic(status ~ distance_to_tertiary_road+
distance_to_city+
distance_to_town+
is_urban+
usage_capacity+
water_source_clean+
water_point_population+
local_population_1km,
data = Osun_wp_sp,
bw = bw.fixed.sig,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413 gwlr.fixed.sig ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-18 00:24:15
Call:
ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city +
distance_to_town + is_urban + usage_capacity + water_source_clean +
water_point_population + local_population_1km, data = Osun_wp_sp,
bw = bw.fixed.sig, family = "binomial", kernel = "gaussian",
adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-129.368 -1.750 1.074 1.742 34.126
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.540e-01 1.055e-01 3.354 0.000796
distance_to_tertiary_road 1.001e-04 2.040e-05 4.910 9.13e-07
distance_to_city -1.764e-05 3.391e-06 -5.202 1.97e-07
distance_to_town -1.544e-05 2.825e-06 -5.466 4.60e-08
is_urbanTRUE -2.667e-01 7.474e-02 -3.569 0.000358
usage_capacity1000 -6.206e-01 6.966e-02 -8.908 < 2e-16
water_source_cleanProtected Shallow Well 4.947e-01 8.496e-02 5.823 5.79e-09
water_source_cleanProtected Spring 1.279e+00 4.384e-01 2.917 0.003530
water_point_population -5.098e-04 4.476e-05 -11.390 < 2e-16
local_population_1km 3.452e-04 1.779e-05 19.407 < 2e-16
Intercept ***
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
is_urbanTRUE ***
usage_capacity1000 ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
water_point_population ***
local_population_1km ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.9 on 4746 degrees of freedom
AIC: 5708.9
Number of Fisher Scoring iterations: 5
AICc: 5708.923
Pseudo R-square value: 0.129406
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2377.371
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -3.7021e+02 -4.3797e+00 3.5590e+00
distance_to_tertiary_road -3.1622e-02 -4.5462e-04 9.1291e-05
distance_to_city -5.4555e-02 -6.5623e-04 -1.3507e-04
distance_to_town -8.6549e-03 -5.2754e-04 -1.6785e-04
is_urbanTRUE -7.3554e+02 -3.4675e+00 -1.6596e+00
usage_capacity1000 -5.5889e+01 -1.0347e+00 -4.1960e-01
water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01 6.2378e-01
water_source_cleanProtected.Spring -1.3630e+03 -5.3436e+00 2.7714e+00
water_point_population -2.9696e-02 -2.2705e-03 -1.2277e-03
local_population_1km -7.7730e-02 4.4281e-04 1.0548e-03
3rd Qu. Max.
Intercept 1.3755e+01 2171.6373
distance_to_tertiary_road 6.3011e-04 0.0237
distance_to_city 1.5921e-04 0.0162
distance_to_town 2.4490e-04 0.0179
is_urbanTRUE 1.0554e+00 995.1840
usage_capacity1000 3.9113e-01 9.2449
water_source_cleanProtected.Shallow.Well 1.9564e+00 66.8914
water_source_cleanProtected.Spring 7.0805e+00 208.3749
water_point_population 4.5879e-04 0.0765
local_population_1km 1.8479e-03 0.0333
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2815.659
AIC : 4418.776
AICc : 4744.213
Pseudo R-square value: 0.5691072
***********************************************************************
Program stops at: 2022-12-18 00:25:19 From the results, we can see that the Geographically Weighted Regression model has a lower AIC (i.e. 4418.776) compared to the Generalised Linear Regression (AIC = 5708.9). This tells us that Geographically Weighted Regression model has improved explainability.
9.3 Model Assessment
We will first convert the model into SFD object as data.frame using the following code chunk.
gwr.fixed.sig <- as.data.frame(gwlr.fixed.sig$SDF)Next, we will label the yhat values (i.e. predicted probability) greater or equal to 0.5 into 1 and else 0. The result of the logic comparison operation will be saved into a field called most.
gwr.fixed.sig <- gwr.fixed.sig %>%
mutate(most = ifelse(
gwr.fixed.sig$yhat >= 0.5, T, F))We will use confusionMatrix() from caret package to generate the confusion matrix.
gwr.fixed.sig$y <- as.factor(gwr.fixed.sig$y)
gwr.fixed.sig$most <- as.factor(gwr.fixed.sig$most)
CM <- confusionMatrix(data=gwr.fixed.sig$most, reference = gwr.fixed.sig$y)
CMConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1833 268
TRUE 281 2374
Accuracy : 0.8846
95% CI : (0.8751, 0.8935)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7661
Mcnemar's Test P-Value : 0.6085
Sensitivity : 0.8671
Specificity : 0.8986
Pos Pred Value : 0.8724
Neg Pred Value : 0.8942
Prevalence : 0.4445
Detection Rate : 0.3854
Detection Prevalence : 0.4418
Balanced Accuracy : 0.8828
'Positive' Class : FALSE
When we compare the overall accuracy is now improved to 88.46% (geographically weighted) compared to the 67.26% that we obtained initially in the non-geographically weighted logistic regression model. In addition, we also noted that both sensitivity and specificity increased significantly from 71.88% to 86.71% and 61.49% to 89.86% respectively. This implies we should apply a local strategy (i.e. gwLR - looking at surrounding neighbours) instead of a global strategy to understand the factors for water points being functional or non-functional.
9.4 Visualising gwLR
The following code chunk below is used to create an interactive point symbol map.
gwr_sf.fixed.sig <- cbind(Osun_wp_sf_selected, gwr.fixed.sig)
tmap_mode("view")tmap mode set to interactive viewingprob_T <- tm_shape(Osun)+
tm_polygons(alpha = 0.1)+
tm_shape(gwr_sf.fixed.sig)+
tm_dots(col = "yhat",
border.col = "gray60",
border.lwd = 1)+
tm_view(set.zoom.limits = c(8,14))
prob_TLikewise, we will visualise how the standard error of coefficient and t-value for the field distance_to_tertiary_road differs for the local models obtained for each state in Osun.
tertiary_TV <- tm_shape(Osun)+
tm_polygons(alpha=0.1)+
tm_shape(gwr_sf.fixed.sig)+
tm_dots(col="distance_to_tertiary_road_TV",
border.col="gray60",
border.lwd = 1)+
tm_view(set.zoom.limits = c(8,14))
tertiary_SE <- tm_shape(Osun)+
tm_polygons(alpha=0.1)+
tm_shape(gwr_sf.fixed.sig)+
tm_dots(col="distance_to_tertiary_road_SE",
border.col="gray60",
border.lwd = 1)+
tm_view(set.zoom.limits = c(8,14))
tmap_arrange(tertiary_SE, tertiary_TV, asp=1, ncol=2, sync=TRUE)Variable(s) "distance_to_tertiary_road_TV" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.With reference to the plot for standard error of coefficient, we can see that standard error of coefficient is generally low for all states (as indicated by pale yellow dots). However, we can see that for several water points in Aiyadade, the local model has a high standard error of coefficient for distance_to_tertiary_road. We can also observe from Section 8.4 and the plot here, that the gwLR model generated in section 8 and section 9 gives different local performance.